This now seems to be permanently on, however there is sufficient coolant, no leaks and not overheating. So presumably a wiring problem or failed sensor or could it be something else? Could anyone advise how to test the sensor please?

Given the MOT is due soon I'm guessing this would count as a failure...

Most of the problems ive found with the coolant level sensor is corrosion building up on the spade
contacts of the sensor and 2 pin cable plug connections. Corrosion builds up due to temperature changes, attracting condensation and oxidises the contacts. (turns green/blue) Non-covered spade contacts are quite prone to this in those situations.
Try, removing the sensor, clean the electrical contacts and probe tips, also the female sockets on the cable plug. Check continuity with a meter between sensor tip and spade connection, also cables on the plug. Add a very small amount of vaseline to the spade contacts when refitting.
Before refitting the sensor, try wetting your thumb and finger and hold the sensor tips. You may get the dash light to flicker.

I'll second prm-comp's comment as that's a more in depth version of what I had to do on mine to resolve the light being on erroneously. I also had to put a cable tie around the plug to hold it onto the sensor to stop it flickering even when all the connections were clean and tight.

Over on the XM club it is often thought that oil in the coolant stops the probes working, making them non-conductive and the thinking is to use proper blue glycol coolant, not never long life OAT types. I presume this transfers over to BX design in a similar way.